Given a family $\mathcal{H}$ of graphs, a graph $G$ is called $\mathcal{H}$-universal if $G$ contains every graph of $\mathcal{H}$ as a subgraph. Following the extensive research on universal graphs of small size for bounded-degree graphs, Alon asked what is the minimum number of edges that a graph must have to be universal for the class of all $n$-vertex graphs that are $D$-degenerate. In this paper, we answer this question up to a factor that is polylogarithmic in $n.$

The last two decades have witnessed a growing trend towards proving sparse random analogues of combinatorial theorems. One unified approach to proving such theorems, formalised by Conlon and Gowers [Ann. of Math. 2016], involves establishing a 'transference principle' which allows one to translate between robust properties in the dense setting and the sparse p-random setting, provided p is not too small. Our results provide a more general transference theorem, extending the results of Conlon and Gowers and also those of Schacht [Ann. of Math. 2016]. Among a variety of other applications, we use this to obtain a sparse counting lemma for graphs and hypergraphs which are not necessarily strictly balanced. Our method achieves asymptotically optimal bounds on the probability p, and the probability of success.

In 1967, Gerencser and Gyárfás determined the exact values of the two-colour Ramsey numbers of paths. In a footnote, they made the following observation: Every 2-edge-coloured complete graph contains a Hamilton path with at most one colour change. Later, this led to a challenging and still wide open conjecture about covering edge-coloured complete graphs with monochromatic paths. Inspired by the original statement, we study paths and cycles with few colour changes in 3-edge-coloured complete graphs. For this, we introduce a new Ramsey-type parameter: For q,k ∈ℕ and a graph G, let R_q^k(G) denote the smallest N ∈ℕ such that every q-edge-coloured complete graph on N vertices contains a copy of G with at most k vertices that are incident to edges in G of different colours. For paths, we show that R_3^1(P_n) = 3n/2 + O(1), and for even cycles, we show that R_3^2(C_n) = 3n/2 + o(n).

The celebrated Bollobás-Eldridge-Catlin packing conjecture states that every n-vertex graph G with minimum degree at least (1-1/Δ+1) n contains every n-vertex graph H of maximum degree at most Δ. Despite considerable attention, the conjecture remains widely open. We show that for bipartite H this threshold can be greatly improved: there is an absolute constant c>0 such that every n-vertex graph G with minimum degree at least (1-c/Δ)n contains every n-vertex bipartite graph H of maximum degree at most Δ, provided Δ is not too large compared to n. Moreover, we prove that this logarithmic improvement is best possible up to the value of the constant.

A graph Γ is said to be universal for a class of graphs H if Γ contains a copy of every H ε H as a subgraph. The number of edges required for a host graph Γ to be universal for the class of D-degenerate graphs on n vertices has been shown to be O((log n)2/D(log log n)5n2-1/D). We generalise this result to r-uniform hypergraphs, showing the following. Given D, r ≥ 2 and n sufficiently large, there exists a constant C = C(D, r) such that there exists a graph with at mostCnr−1/D(log n)2/D(log log n)2r+1edges, which is universal for the class of D-degenerate r-uniform hypergraphs on n vertices. This is tight up to the multiplicative constant and polylogarithmic term.

A graph Γ is said to be universal for a class of graphs ℋ if Γ contains a copy of every H ∈ℋ as a subgraph. The number of edges required for a host graph Γ to be universal for the class of D-degenerate graphs on n vertices has been shown to be O(n^2-1/D(log n)^2/D(loglog n)^5). We generalise this result to r-uniform hypergraphs, showing the following. Given D, r ≥ 2 and n sufficiently large, there exists a constant C = C(D, r) such that there exists a graph with at most Cn^r-1/D(log n)^2/D(loglog n)^2r+1 edges which is universal for the class of D-degenerate r-uniform hypergraphs on n vertices. This is tight up to the polylogarithmic term.

The blow-up lemma states that a system of super-regular pairs contains all bounded degree spanning graphs as subgraphs that embed into a corresponding system of complete pairs. This lemma has far-reaching applications in extremal combinatorics. We prove sparse analogues of the blow-up lemma for subgraphs of random and of pseudorandom graphs. Our main results are the following three sparse versions of the blow-up lemma: one for embedding spanning graphs with maximum degree triangle in subgraphs of G(n, p) with p = C(logn/n)(1/triangle); one for embedding spanning graphs with maximum degree triangle and degeneracy D in subgraphs of G(n, p) with p = C(logn/n)(1/(2D+1)); and one for embedding spanning graphs with maximum degree triangle in (p, cp(max(4,(3 triangle+1)/2))n)-bijumbled graphs. We also consider various applications of these lemmas.

We prove a robust version of a graph embedding theorem of Sauer and Spencer. To state this sparser analogue, we define G(p) to be a random subgraph of G obtained by retaining each edge of G independently with probability p ∈ [0,1], and let m_1(H) be the maximum 1-density of a graph H. We show that for any constant Δ and γ> 0, if G is an n-vertex host graph with minimum degree δ(G) ≥ (1 - 1/2Δ+ γ)n and H is an n-vertex graph with maximum degree Δ(H) ≤ Δ, then for p ≥ Cn^-1/m_1(H)log n, the random subgraph G(p) contains a copy of H with high probability. Our value for p is optimal up to a log-factor. In fact, we prove this result for a more general minimum degree condition on G, by introducing an extension threshold δ_ e(Δ), such that the above result holds for graphs G with δ(G) ≥ (δ_ e(Δ) + γ)n. We show that δ_ e(Δ) ≤ (2Δ-1)/2Δ, and further conjecture that δ_ e(Δ) equals Δ/(Δ+1), which matches the minimum degree condition on G in the Bollobás-Eldridge-Catlin Conjecture. A main tool in our proof is a vertex-spread version of the blow-up lemma of Allen, Böttcher, Hàn, Kohayakawa, and Person, which we believe to be of independent interest.

For a graph $G$ and $p\in[0,1]$, we denote by $G_p$ the random sparsification of $G$ obtained by keeping each edge of $G$ independently, with probability $p$. We show that there exists a $C>0$ such that if $p\geq C(\log n)^{1/3}n^{-2/3}$ and $G$ is an $n$-vertex graph with $n\in 3\mathbb{N}$ and $\delta(G)\geq \tfrac{2n}{3}$, then with high probability $G_p$ contains a triangle factor. Both the minimum degree condition and the probability condition, up to the choice of $C$, are tight. Our result can be viewed as a common strengthening of the seminal theorems of Corr\'adi and Hajnal, which deals with the extremal minimum degree condition for containing triangle factors (corresponding to $p=1$ in our result), and Johansson, Kahn and Vu, which deals with the threshold for the appearance of a triangle factor in $G(n,p)$ (corresponding to $G=K_n$ in our result). It also implies a lower bound on the number of triangle factors in graphs with minimum degree at least $\tfrac{2n}{3}$ which gets close to the truth.

We investigate the appearance of the square of a Hamilton cycle in the model of randomly perturbed graphs, which is, for a given α ∈ ( 0 , 1 ) $$ \alpha \in \left(0,1\right) $$ , the union of any n $$ n $$ -vertex graph with minimum degree α n $$ \alpha n $$ and the binomial random graph G ( n , p ) $$ G\left(n,p\right) $$ . This is known when α > 1 / 2 $$ \alpha >1/2 $$ and we determine the exact perturbed threshold probability in all the remaining cases, that is, for each α ≤ 1 / 2 $$ \alpha \le 1/2 $$ . We demonstrate that, as α $$ \alpha $$ ranges over the interval ( 0 , 1 ) $$ \left(0,1\right) $$ , the threshold performs a countably infinite number of ‘jumps’. Our result has implications on the perturbed threshold for two-universality, where we also fully address all open cases.

We investigate the appearance of the square of a Hamilton cycle in the model of randomly perturbed graphs, which is, for a given $\alpha \in (0,1)$, the union of any $n$-vertex graph with minimum degree $\alpha n$ and the binomial random graph $G(n,p)$. This is known when $\alpha > 1/2$, and we determine the exact perturbed threshold probability in all the remaining cases, i.e., for each $\alpha \le 1/2$. We demonstrate that, as $\alpha$ ranges over the interval $(0,1)$, the threshold performs a countably infinite number of `jumps'. Our result has implications on the perturbed threshold for $2$-universality, where we also fully address all open cases.

Lehel conjectured in the 1970s that the vertices of every red and blue edge-coloured complete graph can be partitioned into two monochromatic cycles. This was confirmed in 2010 by Bessy and Thomassé. However, the host graph G does not have to be complete. It suffices to require that G has minimum degree at least 3n/4, where n is the order of G, as was shown recently by Letzter, confirming a conjecture of Balogh, Barát, Gerbner, Gyárfás and Sárközy. This degree condition is tight. Here we continue this line of research, by proving that for every red and blue edge-colouring of an n-vertex graph of minimum degree at least 2n/3+o(n), there is a partition of the vertex set into three monochromatic cycles. This approximately verifies a conjecture of Pokrovskiy and is essentially tight.

Answering a question by Letzter and Snyder, we prove that for large enough $k$ any $n$-vertex graph $G$ with minimum degree at least $\frac{1}{2k-1}n$ and without odd cycles of length less than $2k+1$ is $3$-colourable. In fact, we prove a stronger result that works with a slightly smaller minimum degree.

We study the problem of finding pairwise vertex-disjoint triangles in the randomly perturbed graphmodel, which is the union of any n-vertex graph G satisfying a given minimum degree condition and the binomial random graph G(n, p). We prove that asymptotically almost surely G boolean OR G(n, p) contains at least min{delta(G), [n/3]} pairwise vertex-disjoint triangles, provided p >= C log n/n, where C is a large enough constant. This is a perturbed version of an old result of Dirac. Our result is asymptotically optimal and answers a question of Han, Morris, and Treglown [RSA, 2021, no. 3, 480-516] in a strong form. We also prove a stability version of our result, which in the case of pairwise vertex-disjoint triangles extends a result of Han, Morris, and Treglown [RSA, 2021, no. 3, 480-516]. Together with a result of Balogh, Treglown, and Wagner [CPC, 2019, no. 2, 159-176], this fully resolves the existence of triangle factors in randomly perturbed graphs. We believe that the methods introduced in this paper are useful for a variety of related problems: we discuss possible generalisations to clique factors, cycle factors, and 2-universality.

Packing problems in combinatorics concern the edge disjoint embedding of a family of guest (hyper)graphs into a given host (hyper)graph. Questions of this type are intimately connected to the field of design theory, and have a variety of significant applications. The area has seen important progress in the last two decades, with a number of powerful new methods developed. Here, I will survey some major results contributing to this progress, alongside background, and some ideas concerning the methods involved.

We consider the following question. When is the random $k$-uniform hypergraph $\Gamma=G^{(k)}(N,p)$ likely to be $r$-partition universal for $k$-uniform hypergraphs of bounded degree and degeneracy? That is, for which~$p$ can we guarantee asymptotically almost surely that in any $r$-colouring of $E(\Gamma)$ there exists a colour $\chi$ such that in $\Gamma$ there are $\chi$-monochromatic copies of all $k$-uniform hypergraphs of maximum vertex degree $\Delta$, degeneracy at most $D$, and $cN$ vertices for some constant $c=c(D,\Delta)>0$. We show that if $\mu>0$ is fixed, then $p\ge N^{-1/D+\mu}$ suffices for a positive answer if $N$ is large. On the other hand, for $p=o(N^{-1/D})$ we show that $G^{(k)}(N,p)$ is likely not to contain some graphs of maximum degree $\Delta$ and degeneracy $D$ on $cN$ vertices at all. This improves the best upper bounds on the minimum number of edges required for a $k$-uniform hypergraph to be partition universal (even for $k=2$) and also for the size-Ramsey problem for most $k$-uniform hypergraphs of bounded degree and degeneracy.

The bandwidth theorem [Mathematische Annalen, 343(1):175--205, 2009] states that any $n$-vertex graph $G$ with minimum degree $(\frac{k-1}{k}+o(1))n$ contains all $n$-vertex $k$-colourable graphs $H$ with bounded maximum degree and bandwidth $o(n)$. In [arXiv:1612.00661] a random graph analogue of this statement is proved: for $p\gg (\frac{\log n}{n})^{1/\Delta}$ a.a.s. each spanning subgraph $G$ of $G(n,p)$ with minimum degree $(\frac{k-1}{k}+o(1))pn$ contains all $n$-vertex $k$-colourable graphs $H$ with maximum degree $\Delta$, bandwidth $o(n)$, and at least $C p^{-2}$ vertices not contained in any triangle. This restriction on vertices in triangles is necessary, but limiting. In this paper we consider how it can be avoided. A special case of our main result is that, under the same conditions, if additionally all vertex neighbourhoods in $G$ contain many copies of $K_\Delta$ then we can drop the restriction on $H$ that $Cp^{-2}$ vertices should not be in triangles.

We prove that one can perfectly pack degenerate graphs into complete or dense n-vertex quasirandom graphs, provided that all the degenerate graphs have maximum degree $$o\left( {{n \over {\log \,n}}} \right)$$ , and in addition Ω(n) of them have at most (1 − Ω(1))n vertices and Ω(n) leaves. This proves Ringel’s conjecture and the Gyárfás Tree Packing Conjecture for all but an exponentially small fraction of trees (or sequences of trees, respectively).

We obtain an approximate sparse hypergraph version of the blow-up lemma, showing that partite hypergraphs with sufficient regularity of small subgraph counts behave as if they were complete partite for the purpose of embedding bounded degree hypergraphs.